{"id":3829,"date":"2023-05-31T15:31:52","date_gmt":"2023-05-31T15:31:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3829"},"modified":"2024-10-18T20:58:36","modified_gmt":"2024-10-18T20:58:36","slug":"voting-theory-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/voting-theory-fresh-take\/","title":{"raw":"Voting Theory: Fresh Take","rendered":"Voting Theory: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Determine the winner and assess the fairness of an election using preference ballots<\/li>\r\n\t<li>Determine the winner and assess the fairness of an election using the Instant Runoff method<\/li>\r\n\t<li>Determine the winner and assess the fairness of an election using a Borda count<\/li>\r\n\t<li>Determine the winner and assess the fairness of an election using Copeland's method<\/li>\r\n\t<li>Determine the winner and assess the fairness of an election using the Approval Voting method<\/li>\r\n\t<li>Apply Arrow\u2019s Impossibility Theorem<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Ever been in a situation where you and your friends can't decide on a movie to watch? Or maybe you've wondered how elections work in a democratic country? Welcome to Voting Theory! This section introduces you to various methods of voting, each with its own pros and cons. From Preference Ballots that let you rank your choices, to different voting methods like Instant Runoff, Borda Count, Copeland's Method, and Approval Voting, we explore how to make group decisions that are as fair as possible.<\/p>\r\n<h2>Preference Ballot<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <br \/>\r\n<br \/>\r\n<strong>Preference Ballot:<\/strong> This isn't your usual 'pick one' ballot. Here, you rank your choices, giving a fuller picture of what you really want. <br \/>\r\n<br \/>\r\n<strong>Preference Schedule:<\/strong> Think of it as a cheat sheet that organizes everyone's rankings, making it easier to see group preferences.<\/div>\r\n<p>The following video will give you a summary of what issues can arise from elections, as well as how a preference schedule is used in elections.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6rhpq1ozmuQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/(New+Version+Available)+Introduction+to+Voting+Theory+and+Preference+Tables.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c(New Version Available) Introduction to Voting Theory and Preference Tables\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7747[\/ohm2_question]<\/section>\r\n<h2>Plurality<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <br \/>\r\n<br \/>\r\nIn the realm of voting, the <strong>plurality method<\/strong> and the <strong>Condorcet winner<\/strong> are two key concepts that often come into play. The plurality method is the most straightforward: the candidate with the most first-choice votes wins. However, this method doesn't always reflect the majority's preference, as a candidate can win with a plurality but not a majority of votes. On the other hand, the Condorcet winner is a candidate who would win in every one-to-one comparison against all other candidates. This concept introduces a fairness criterion, aiming to identify a candidate that truly represents the majority's preference. <br \/>\r\n<br \/>\r\n<strong>Plurality Method:<\/strong> Focus only on the first-choice votes. The candidate with the most first-choice votes wins, even if they don't have an absolute majority. <br \/>\r\n<br \/>\r\n<strong>Condorcet Winner:<\/strong> This is the candidate who would beat every other candidate in a one-on-one vote. To find the Condorcet Winner, compare each candidate against all others in one-to-one matchups and see who comes out on top in each.<\/div>\r\n<section class=\"textbox example\">Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)[footnote]This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See https:\/\/www.piercecountywa.gov\/DocumentCenter\/View\/6985\/summary?bidId=[\/footnote]\u00a0The voting schedule is shown below. Is there a Condorcet winner in this election?\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">[latex]44[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]14[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]20[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]70[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]22[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]80[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]44[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">1st choice<\/td>\r\n<td style=\"text-align: center;\">G<\/td>\r\n<td style=\"text-align: center;\">G<\/td>\r\n<td style=\"text-align: center;\">G<\/td>\r\n<td style=\"text-align: center;\">M<\/td>\r\n<td style=\"text-align: center;\">M<\/td>\r\n<td style=\"text-align: center;\">B<\/td>\r\n<td style=\"text-align: center;\">B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">2nd choice<\/td>\r\n<td style=\"text-align: center;\">M<\/td>\r\n<td style=\"text-align: center;\">B<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">G<\/td>\r\n<td style=\"text-align: center;\">B<\/td>\r\n<td style=\"text-align: center;\">M<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">3rd choice<\/td>\r\n<td style=\"text-align: center;\">B<\/td>\r\n<td style=\"text-align: center;\">M<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">B<\/td>\r\n<td style=\"text-align: center;\">G<\/td>\r\n<td style=\"text-align: center;\">G<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Note: In the third column and last column, those voters only recorded a first-place vote, so we don\u2019t know who their second and third choices would have been.<br \/>\r\n[reveal-answer q=\"4331\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4331\"]Using plurality method:<\/p>\r\n<p>Determining the Condorcet Winner:<\/p>\r\n<p>G vs M: [latex]44+14+20=78[\/latex] prefer G, [latex]70+22+80=172[\/latex] prefer M: M preferred<\/p>\r\n<p>G vs B: [latex]44+14+20+70=148[\/latex] prefer G, [latex]22+80+39=141[\/latex] prefer B: G preferred<\/p>\r\n<p>M vs B: [latex]44+70+22=136[\/latex] prefer M, [latex]14+80+39=133[\/latex] prefer B: M preferred<\/p>\r\n<p>M is the Condorcet winner, based on the information we have.<br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The following video gives another mini lesson that covers the plurality method of voting as well as the idea of a Condorcet Winner.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/r-VmxJQFMq8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Plurality+Method+and+Condorcet+Criterion.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Plurality Method and Condorcet Criterion\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Instant Runoff Voting<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <br \/>\r\n<br \/>\r\n<strong>Instant Runoff Voting (IRV)<\/strong>, also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting. <br \/>\r\n<br \/>\r\nIRV involves preference ballots and a process of eliminating candidates with the fewest first-place votes. Votes for the eliminated candidate are redistributed to voters' next choices. This process continues until a choice has a majority (over [latex]50\\%[\/latex]).<br \/>\r\n<br \/>\r\n<strong>Monotonicity criterion<\/strong> refers to the principle that if voters change their votes to increase the preference for a candidate, it should not harm that candidate's chances of winning.<br \/>\r\n<br \/>\r\nThe criterion is violated in some elections, but this doesn't mean IRV always violates it; it depends on the election context.<\/div>\r\n<p>Here is an overview video that provides the definition of IRV, as well as an example of how to determine the winner of an election using IRV.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>https:\/\/youtu.be\/6axH6pcuyhQ<\/p>\r\n<p><strong>Please note:<\/strong>\u00a0at 2:50 in the video it says [latex]9+2+8=18[\/latex], it should say [latex]9+2+8=19[\/latex], so [latex]D=19[\/latex].<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Voting+Theory_+Instant+Runoff+Voting.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Instant Runoff Voting\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider again this election. Find the winner using IRV.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]70[\/latex]<\/td>\r\n<td>[latex]22[\/latex]<\/td>\r\n<td>[latex]80[\/latex]<\/td>\r\n<td>[latex]39[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>M<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>G<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>B<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[reveal-answer q=\"954410\"]Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"954410\"]<\/p>\r\n<p>G has the fewest first-choice votes, and so is eliminated first. The [latex]44[\/latex] voters who listed M as the second choice go to McCarthy. The [latex]14[\/latex] voters who listed B as second choice go to Bunney. The [latex]20[\/latex] voters who did not list a second choice do not get transferred. They simply get eliminated.<\/p>\r\n<p>McCarthy gets [latex]92 + 44 =\u00a0 136[\/latex]; Bunney gets [latex]119 + 14 = 133[\/latex]<\/p>\r\n<p>We are down to two possibilities with McCarthy at [latex]136[\/latex] and Bunney at [latex]133[\/latex]. McCarthy is declared the winner.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Borda Count<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <br \/>\r\n<br \/>\r\nThe <strong>Borda count<\/strong> method adds a new layer to voting by assigning points to candidates based on their ranking. While this system can sometimes yield a more \"consensus-based\" winner, it has its drawbacks. For instance, a candidate with a majority of first-place votes might not win, violating the <strong>majority criterion<\/strong>. This method is often used in sports awards and other scenarios where a more nuanced view of preferences is beneficial. <br \/>\r\n<br \/>\r\n<strong>Borda Count:<\/strong> In this method, each rank you give to a candidate earns them points. The candidate with the most points wins, but this may not always align with the majority's first-choice preference. <br \/>\r\n<br \/>\r\n<strong>Majority Criterion:<\/strong> If a candidate has a majority of first-place votes, they should ideally be the winner. Borda Count can sometimes violate this criterion. <br \/>\r\n<br \/>\r\n<strong>Consensus-Based Voting:<\/strong> Borda Count aims for a more broadly acceptable option rather than focusing solely on first-choice votes. It considers every voter's entire ranking to determine the outcome.<\/div>\r\n<section class=\"textbox example\">Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)[footnote]This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See https:\/\/www.piercecountywa.gov\/DocumentCenter\/View\/6985\/summary?bidId=[\/footnote]\u00a0The voting schedule is shown below. Find the winner using Borda count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate [latex]1[\/latex] point, the points they would normally get for last place.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]44[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]70[\/latex]<\/td>\r\n<td>[latex]22[\/latex]<\/td>\r\n<td>[latex]80[\/latex]<\/td>\r\n<td>[latex]39[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>M<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>M<\/td>\r\n<td>B<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>G<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>B<\/td>\r\n<td>M<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>B<\/td>\r\n<td>G<\/td>\r\n<td>G<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[reveal-answer q=\"159776\"]Solution[\/reveal-answer] [hidden-answer a=\"159776\"]\r\n\r\n<p>We give [latex]1[\/latex] point for 3rd place, [latex]2[\/latex] points for 2nd place, and [latex]3[\/latex] points for 1st place. In the columns where voters only listed first place choice, the same number of votes are awarded to each of the other candidates as 3rd place vote multipliers.<\/p>\r\n<p>Column 1: [latex]44[\/latex] voters<\/p>\r\n<p>Goings (G):[latex]44\u00d73=132[\/latex] points<\/p>\r\n<p>McCarthy (M): [latex]44\u00d72=88[\/latex] points<\/p>\r\n<p>Bunney (B):[latex]44\u00d71=44[\/latex] points<\/p>\r\n<p>Column 2: [latex]14[\/latex] voters<\/p>\r\n<p>Goings (G): [latex]14\u00d73=42[\/latex] points<\/p>\r\n<p>Bunney (B): [latex]14\u00d72=28[\/latex] points<\/p>\r\n<p>McCarthy (M): [latex]14\u00d71=14[\/latex] points<\/p>\r\n<p>Column 3: [latex]20[\/latex] voters<\/p>\r\n<p>Goings (G): [latex]20\u00d73=60[\/latex] points<\/p>\r\n<p>McCarthy (M): [latex]20\u00d71=20[\/latex] points<\/p>\r\n<p>Bunney (B): [latex]20\u00d71=20[\/latex] points<\/p>\r\n<p>Column 4: [latex]70[\/latex] voters<\/p>\r\n<p>McCarthy (M): [latex]70\u00d73=210[\/latex] points<\/p>\r\n<p>Goings (G): [latex]70\u00d72=140[\/latex] points<\/p>\r\n<p>Bunney (B): [latex]70\u00d71=70[\/latex] points<\/p>\r\n<p>Column 5: [latex]22[\/latex] voters<\/p>\r\n<p>McCarthy (M): [latex]22\u00d73=66[\/latex] points<\/p>\r\n<p>Bunney (B): [latex]22\u00d72=44[\/latex] points<\/p>\r\n<p>Goings (G): [latex]22\u00d71=22[\/latex] points<\/p>\r\n<p>Column 6: [latex]80[\/latex] voters<\/p>\r\n<p>Bunney (B): [latex]80\u00d73=240[\/latex] points<\/p>\r\n<p>McCarthy (M): [latex]80\u00d72=160[\/latex] points<\/p>\r\n<p>Goings (G): [latex]80\u00d71=80[\/latex] points<\/p>\r\n<p>Column 7: [latex]39[\/latex] voters<\/p>\r\n<p>Bunney (B): [latex]39\u00d73=117[\/latex] points<\/p>\r\n<p>McCarthy (M): [latex]39\u00d71=39[\/latex] points<\/p>\r\n<p>Goings (G): [latex]39\u00d71=39[\/latex] points<\/p>\r\n<p>G: [latex]132+42+60+140+22+80+39 = 515[\/latex] points<\/p>\r\n<p>M: [latex]88+14+20+210+66+160+39 = 597[\/latex] points<\/p>\r\n<p>B: [latex]44+28+20+70+44+240+117 = 563[\/latex] points<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>McCarthy would be the winner using Borda Count.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>Watch the following for more information on the Borda count method.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hVG9jmA4FBU?si=hLejTbSwcVLcXA9x\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Borda+count.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Borda count\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Copeland\u2019s Method<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <br \/>\r\n<br \/>\r\n<strong>Copeland's Method:<\/strong> A voting method that aims to satisfy the Condorcet Criterion by conducting pairwise comparisons between candidates. The more preferred candidate in each pair gets [latex]1[\/latex] point, and in case of a tie, each gets [latex]\\frac{1}{2}[\/latex] point. The candidate with the most points wins. <br \/>\r\n<br \/>\r\n<strong>IIA Criterion:<\/strong> Removing a non-winning choice from the ballot should not change the election's winner. <br \/>\r\n<br \/>\r\nThings to consider when applying Copeland's method:\r\n\r\n<ul>\r\n\t<li><strong>Pairwise Comparisons:<\/strong> Always compare candidates in pairs to determine the more preferred one.<\/li>\r\n\t<li><strong>Point Allocation:<\/strong> Award [latex]1[\/latex] point to the more preferred candidate and [latex]\\frac{1}{2}[\/latex] point to each in case of a tie.<\/li>\r\n\t<li><strong>Totaling Points:<\/strong> Sum up the points for each candidate; the one with the most points wins.<\/li>\r\n\t<li><strong>Check for IIA:<\/strong> Ensure that removing a non-winning candidate doesn't change the election outcome.<\/li>\r\n\t<li><strong>Tie-Breaking:<\/strong> Copeland's method can often result in ties, requiring more advanced methods for resolution.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Consider the preference schedule below, in which a company\u2019s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity. Determine the winner using Copeland\u2019s method.<\/p>\r\n<p>Initial votes :<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1st choice<\/td>\r\n<td>B<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>E<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2nd choice<\/td>\r\n<td>C<\/td>\r\n<td>A<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>E<\/td>\r\n<td>A<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3rd choice<\/td>\r\n<td>A<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<td>A<\/td>\r\n<td>A<\/td>\r\n<td>D<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4th choice<\/td>\r\n<td>D<\/td>\r\n<td>B<\/td>\r\n<td>A<\/td>\r\n<td>E<\/td>\r\n<td>C<\/td>\r\n<td>B<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5th choice<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>E<\/td>\r\n<td>B<\/td>\r\n<td>D<\/td>\r\n<td>C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[reveal-answer q=\"944614\"]Show Solution[\/reveal-answer] [hidden-answer a=\"944614\"] With [latex]5[\/latex] candidates, there are [latex]10[\/latex] comparisons to make:\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A vs B: [latex]11[\/latex] votes to [latex]9[\/latex] votes<\/td>\r\n<td>A gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs C: [latex]3[\/latex] votes to [latex]17[\/latex] votes<\/td>\r\n<td>C gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs D: [latex]10[\/latex] votes to [latex]10[\/latex] votes<\/td>\r\n<td>A gets [latex]\\frac{1}{2}[\/latex] point, D gets [latex]\\frac{1}{2}[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A vs E: [latex]17[\/latex] votes to [latex]3[\/latex] votes<\/td>\r\n<td>A gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs C: [latex]10[\/latex] votes to [latex]10[\/latex] votes<\/td>\r\n<td>B gets [latex]\\frac{1}{2}[\/latex] point, C gets [latex]\\frac{1}{2}[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs D: [latex]9[\/latex] votes to [latex]11[\/latex] votes<\/td>\r\n<td>D gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>B vs E: [latex]13[\/latex] votes to [latex]7[\/latex] votes<\/td>\r\n<td>B gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C vs D: [latex]9[\/latex] votes to [latex]11[\/latex] votes<\/td>\r\n<td>D gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>C vs E: [latex]17[\/latex] votes to [latex]3[\/latex] votes<\/td>\r\n<td>C gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D vs E: [latex]17[\/latex] votes to [latex]3[\/latex] votes<\/td>\r\n<td>D gets [latex]1[\/latex] point<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>Totaling these up: A gets [latex]2\\frac{1}{2}[\/latex] points B gets [latex]1\\frac{1}{2}[\/latex] points C gets [latex]2\\frac{1}{2}[\/latex] points D gets [latex]3\\frac{1}{2}[\/latex] points E gets [latex]0[\/latex] points. Using Copeland\u2019s Method, we declare D as the winner. Notice that in this case, D is not a Condorcet Winner. While Copeland\u2019s method will also select a Condorcet Candidate as the winner, the method still works in cases where there is no Condorcet Winner.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>Watch the following for more information on the Copeland's method.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/sh3IEALnpNg?si=7F5a0kyLtxii-VS7\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Copeland's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Copeland's Method\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Approval Voting<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p><strong>Approval Voting:<\/strong> A voting method where you mark all choices you find acceptable. The option with the most approval wins. Approval voting can easily violate the majority criterion, meaning the candidate with the most first-choice votes may not win.<\/p>\r\n<p><strong>Strategic Insincere Voting:<\/strong> This method is susceptible to voters not voting their true preference to increase the chances of their choice winning.<\/p>\r\n\r\nThings to consider when applying the approval voting method:\r\n\r\n<ul>\r\n\t<li><strong>Marking Choices:<\/strong> In approval voting, you're not ranking candidates; you're simply marking all that you find acceptable.<\/li>\r\n\t<li><strong>Tallying Votes:<\/strong> Count the number of approvals each option receives. The one with the most wins.<\/li>\r\n\t<li><strong>Strategic Voting:<\/strong> Some voters may mark options they find less preferable to increase the chances of their true preference winning.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">A group of friends are trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an \u201cX\u201d which movies they are willing to watch. The results are:\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">Bob<\/td>\r\n<td style=\"text-align: center;\">Ann<\/td>\r\n<td style=\"text-align: center;\">Marv<\/td>\r\n<td style=\"text-align: center;\">Alice<\/td>\r\n<td style=\"text-align: center;\">Eve<\/td>\r\n<td style=\"text-align: center;\">Omar<\/td>\r\n<td style=\"text-align: center;\">Lupe<\/td>\r\n<td style=\"text-align: center;\">Dave<\/td>\r\n<td style=\"text-align: center;\">Tish<\/td>\r\n<td style=\"text-align: center;\">Jim<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">Avatar: The Way of Water<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">Spider-Man: Across the Spider-Verse<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">Everything Everywhere All at Once<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">\u00a0<\/td>\r\n<td style=\"text-align: center;\">X<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[reveal-answer q=\"109152\"]Solution[\/reveal-answer] [hidden-answer a=\"109152\"]\r\n\r\n<p>Totaling the results, we find:<\/p>\r\n<p>Avatar: The Way of Water received [latex]5[\/latex] approvals.<\/p>\r\n<p>Spider-Man: Across the Spider-Verse received [latex]6[\/latex] approvals.<\/p>\r\n<p>Everything Everywhere All at Once received [latex]7[\/latex] approvals.<\/p>\r\n<p>In this vote, Everything Everywhere All at Once would be the winner.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>Watch the following for more information on the approval voting method.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vv1pquvAIDI?si=ZYJfXo-dVuna6u5l\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Approval+Voting.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Approval Voting\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Which Method is Fair?<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong>\r\n<p>Which Method is Fair? The search for a perfect voting method is elusive, as no method satisfies all fairness criteria.<\/p>\r\n<p><strong>Arrow's Impossibility Theorem:<\/strong> Proven by Kenneth Arrow in 1949, this theorem states that no voting method can satisfy all fairness criteria.<\/p>\r\n<p><strong>Condorcet's Voting Paradox:<\/strong> This paradox shows that voting preferences are not transitive, meaning that if A is preferred over B, and B over C, it doesn't necessarily mean A is preferred over C.<\/p>\r\n<p><strong>Method Selection:<\/strong> The choice of voting method often depends on what seems most fair for the specific situation.<\/p>\r\n\r\nThings to consider when evaluating voting methods and interpreting results:\r\n\r\n<ul>\r\n\t<li><strong>Fairness Criteria:<\/strong> Always consider the fairness criteria when evaluating a voting method.<\/li>\r\n\t<li><strong>Transitivity:<\/strong> Be cautious of the Condorcet's Voting Paradox when interpreting voting results.<\/li>\r\n\t<li><strong>Method Choice:<\/strong> The \"best\" method may vary depending on the context and what fairness criteria are most important.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>Watch the following for more information on the four fairness criterion and the Arrow's Impossibility Theorem.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/CuOLQT9P11I?si=I0cMmynZfoF7ZEbp\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Fairness+Criterion.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Fairness Criterion\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine the winner and assess the fairness of an election using preference ballots<\/li>\n<li>Determine the winner and assess the fairness of an election using the Instant Runoff method<\/li>\n<li>Determine the winner and assess the fairness of an election using a Borda count<\/li>\n<li>Determine the winner and assess the fairness of an election using Copeland&#8217;s method<\/li>\n<li>Determine the winner and assess the fairness of an election using the Approval Voting method<\/li>\n<li>Apply Arrow\u2019s Impossibility Theorem<\/li>\n<\/ul>\n<\/section>\n<p>Ever been in a situation where you and your friends can&#8217;t decide on a movie to watch? Or maybe you&#8217;ve wondered how elections work in a democratic country? Welcome to Voting Theory! This section introduces you to various methods of voting, each with its own pros and cons. From Preference Ballots that let you rank your choices, to different voting methods like Instant Runoff, Borda Count, Copeland&#8217;s Method, and Approval Voting, we explore how to make group decisions that are as fair as possible.<\/p>\n<h2>Preference Ballot<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <\/p>\n<p><strong>Preference Ballot:<\/strong> This isn&#8217;t your usual &#8216;pick one&#8217; ballot. Here, you rank your choices, giving a fuller picture of what you really want. <\/p>\n<p><strong>Preference Schedule:<\/strong> Think of it as a cheat sheet that organizes everyone&#8217;s rankings, making it easier to see group preferences.<\/div>\n<p>The following video will give you a summary of what issues can arise from elections, as well as how a preference schedule is used in elections.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6rhpq1ozmuQ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/(New+Version+Available)+Introduction+to+Voting+Theory+and+Preference+Tables.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c(New Version Available) Introduction to Voting Theory and Preference Tables\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7747\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7747&theme=lumen&iframe_resize_id=ohm7747&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Plurality<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <\/p>\n<p>In the realm of voting, the <strong>plurality method<\/strong> and the <strong>Condorcet winner<\/strong> are two key concepts that often come into play. The plurality method is the most straightforward: the candidate with the most first-choice votes wins. However, this method doesn&#8217;t always reflect the majority&#8217;s preference, as a candidate can win with a plurality but not a majority of votes. On the other hand, the Condorcet winner is a candidate who would win in every one-to-one comparison against all other candidates. This concept introduces a fairness criterion, aiming to identify a candidate that truly represents the majority&#8217;s preference. <\/p>\n<p><strong>Plurality Method:<\/strong> Focus only on the first-choice votes. The candidate with the most first-choice votes wins, even if they don&#8217;t have an absolute majority. <\/p>\n<p><strong>Condorcet Winner:<\/strong> This is the candidate who would beat every other candidate in a one-on-one vote. To find the Condorcet Winner, compare each candidate against all others in one-to-one matchups and see who comes out on top in each.<\/div>\n<section class=\"textbox example\">Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)<a class=\"footnote\" title=\"This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See https:\/\/www.piercecountywa.gov\/DocumentCenter\/View\/6985\/summary?bidId=\" id=\"return-footnote-3829-1\" href=\"#footnote-3829-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0The voting schedule is shown below. Is there a Condorcet winner in this election?<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">[latex]44[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]14[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]20[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]70[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]22[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]80[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]44[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">1st choice<\/td>\n<td style=\"text-align: center;\">G<\/td>\n<td style=\"text-align: center;\">G<\/td>\n<td style=\"text-align: center;\">G<\/td>\n<td style=\"text-align: center;\">M<\/td>\n<td style=\"text-align: center;\">M<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">2nd choice<\/td>\n<td style=\"text-align: center;\">M<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">G<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">M<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">3rd choice<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">M<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">B<\/td>\n<td style=\"text-align: center;\">G<\/td>\n<td style=\"text-align: center;\">G<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Note: In the third column and last column, those voters only recorded a first-place vote, so we don\u2019t know who their second and third choices would have been.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4331\">Show Solution<\/button><\/p>\n<div id=\"q4331\" class=\"hidden-answer\" style=\"display: none\">Using plurality method:<\/p>\n<p>Determining the Condorcet Winner:<\/p>\n<p>G vs M: [latex]44+14+20=78[\/latex] prefer G, [latex]70+22+80=172[\/latex] prefer M: M preferred<\/p>\n<p>G vs B: [latex]44+14+20+70=148[\/latex] prefer G, [latex]22+80+39=141[\/latex] prefer B: G preferred<\/p>\n<p>M vs B: [latex]44+70+22=136[\/latex] prefer M, [latex]14+80+39=133[\/latex] prefer B: M preferred<\/p>\n<p>M is the Condorcet winner, based on the information we have.\n<\/div>\n<\/div>\n<\/section>\n<p>The following video gives another mini lesson that covers the plurality method of voting as well as the idea of a Condorcet Winner.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/r-VmxJQFMq8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Plurality+Method+and+Condorcet+Criterion.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Plurality Method and Condorcet Criterion\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Instant Runoff Voting<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <\/p>\n<p><strong>Instant Runoff Voting (IRV)<\/strong>, also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting. <\/p>\n<p>IRV involves preference ballots and a process of eliminating candidates with the fewest first-place votes. Votes for the eliminated candidate are redistributed to voters&#8217; next choices. This process continues until a choice has a majority (over [latex]50\\%[\/latex]).<\/p>\n<p><strong>Monotonicity criterion<\/strong> refers to the principle that if voters change their votes to increase the preference for a candidate, it should not harm that candidate&#8217;s chances of winning.<\/p>\n<p>The criterion is violated in some elections, but this doesn&#8217;t mean IRV always violates it; it depends on the election context.<\/p><\/div>\n<p>Here is an overview video that provides the definition of IRV, as well as an example of how to determine the winner of an election using IRV.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Voting Theory: Instant Runoff Voting\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6axH6pcuyhQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><strong>Please note:<\/strong>\u00a0at 2:50 in the video it says [latex]9+2+8=18[\/latex], it should say [latex]9+2+8=19[\/latex], so [latex]D=19[\/latex].<\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Voting+Theory_+Instant+Runoff+Voting.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Instant Runoff Voting\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Consider again this election. Find the winner using IRV.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]44[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<td>[latex]22[\/latex]<\/td>\n<td>[latex]80[\/latex]<\/td>\n<td>[latex]39[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>M<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td>&nbsp;<\/td>\n<td>G<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td>&nbsp;<\/td>\n<td>B<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q954410\">Solution<\/button><\/p>\n<div id=\"q954410\" class=\"hidden-answer\" style=\"display: none\">\n<p>G has the fewest first-choice votes, and so is eliminated first. The [latex]44[\/latex] voters who listed M as the second choice go to McCarthy. The [latex]14[\/latex] voters who listed B as second choice go to Bunney. The [latex]20[\/latex] voters who did not list a second choice do not get transferred. They simply get eliminated.<\/p>\n<p>McCarthy gets [latex]92 + 44 =\u00a0 136[\/latex]; Bunney gets [latex]119 + 14 = 133[\/latex]<\/p>\n<p>We are down to two possibilities with McCarthy at [latex]136[\/latex] and Bunney at [latex]133[\/latex]. McCarthy is declared the winner.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Borda Count<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <\/p>\n<p>The <strong>Borda count<\/strong> method adds a new layer to voting by assigning points to candidates based on their ranking. While this system can sometimes yield a more &#8220;consensus-based&#8221; winner, it has its drawbacks. For instance, a candidate with a majority of first-place votes might not win, violating the <strong>majority criterion<\/strong>. This method is often used in sports awards and other scenarios where a more nuanced view of preferences is beneficial. <\/p>\n<p><strong>Borda Count:<\/strong> In this method, each rank you give to a candidate earns them points. The candidate with the most points wins, but this may not always align with the majority&#8217;s first-choice preference. <\/p>\n<p><strong>Majority Criterion:<\/strong> If a candidate has a majority of first-place votes, they should ideally be the winner. Borda Count can sometimes violate this criterion. <\/p>\n<p><strong>Consensus-Based Voting:<\/strong> Borda Count aims for a more broadly acceptable option rather than focusing solely on first-choice votes. It considers every voter&#8217;s entire ranking to determine the outcome.<\/div>\n<section class=\"textbox example\">Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)<a class=\"footnote\" title=\"This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See https:\/\/www.piercecountywa.gov\/DocumentCenter\/View\/6985\/summary?bidId=\" id=\"return-footnote-3829-2\" href=\"#footnote-3829-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>\u00a0The voting schedule is shown below. Find the winner using Borda count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate [latex]1[\/latex] point, the points they would normally get for last place.<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]44[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<td>[latex]22[\/latex]<\/td>\n<td>[latex]80[\/latex]<\/td>\n<td>[latex]39[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>M<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>M<\/td>\n<td>B<\/td>\n<td>&nbsp;<\/td>\n<td>G<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>B<\/td>\n<td>M<\/td>\n<td>&nbsp;<\/td>\n<td>B<\/td>\n<td>G<\/td>\n<td>G<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q159776\">Solution<\/button> <\/p>\n<div id=\"q159776\" class=\"hidden-answer\" style=\"display: none\">\n<p>We give [latex]1[\/latex] point for 3rd place, [latex]2[\/latex] points for 2nd place, and [latex]3[\/latex] points for 1st place. In the columns where voters only listed first place choice, the same number of votes are awarded to each of the other candidates as 3rd place vote multipliers.<\/p>\n<p>Column 1: [latex]44[\/latex] voters<\/p>\n<p>Goings (G):[latex]44\u00d73=132[\/latex] points<\/p>\n<p>McCarthy (M): [latex]44\u00d72=88[\/latex] points<\/p>\n<p>Bunney (B):[latex]44\u00d71=44[\/latex] points<\/p>\n<p>Column 2: [latex]14[\/latex] voters<\/p>\n<p>Goings (G): [latex]14\u00d73=42[\/latex] points<\/p>\n<p>Bunney (B): [latex]14\u00d72=28[\/latex] points<\/p>\n<p>McCarthy (M): [latex]14\u00d71=14[\/latex] points<\/p>\n<p>Column 3: [latex]20[\/latex] voters<\/p>\n<p>Goings (G): [latex]20\u00d73=60[\/latex] points<\/p>\n<p>McCarthy (M): [latex]20\u00d71=20[\/latex] points<\/p>\n<p>Bunney (B): [latex]20\u00d71=20[\/latex] points<\/p>\n<p>Column 4: [latex]70[\/latex] voters<\/p>\n<p>McCarthy (M): [latex]70\u00d73=210[\/latex] points<\/p>\n<p>Goings (G): [latex]70\u00d72=140[\/latex] points<\/p>\n<p>Bunney (B): [latex]70\u00d71=70[\/latex] points<\/p>\n<p>Column 5: [latex]22[\/latex] voters<\/p>\n<p>McCarthy (M): [latex]22\u00d73=66[\/latex] points<\/p>\n<p>Bunney (B): [latex]22\u00d72=44[\/latex] points<\/p>\n<p>Goings (G): [latex]22\u00d71=22[\/latex] points<\/p>\n<p>Column 6: [latex]80[\/latex] voters<\/p>\n<p>Bunney (B): [latex]80\u00d73=240[\/latex] points<\/p>\n<p>McCarthy (M): [latex]80\u00d72=160[\/latex] points<\/p>\n<p>Goings (G): [latex]80\u00d71=80[\/latex] points<\/p>\n<p>Column 7: [latex]39[\/latex] voters<\/p>\n<p>Bunney (B): [latex]39\u00d73=117[\/latex] points<\/p>\n<p>McCarthy (M): [latex]39\u00d71=39[\/latex] points<\/p>\n<p>Goings (G): [latex]39\u00d71=39[\/latex] points<\/p>\n<p>G: [latex]132+42+60+140+22+80+39 = 515[\/latex] points<\/p>\n<p>M: [latex]88+14+20+210+66+160+39 = 597[\/latex] points<\/p>\n<p>B: [latex]44+28+20+70+44+240+117 = 563[\/latex] points<\/p>\n<p>&nbsp;<\/p>\n<p>McCarthy would be the winner using Borda Count.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following for more information on the Borda count method.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/hVG9jmA4FBU?si=hLejTbSwcVLcXA9x\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Borda+count.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Borda count\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Copeland\u2019s Method<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong> <\/p>\n<p><strong>Copeland&#8217;s Method:<\/strong> A voting method that aims to satisfy the Condorcet Criterion by conducting pairwise comparisons between candidates. The more preferred candidate in each pair gets [latex]1[\/latex] point, and in case of a tie, each gets [latex]\\frac{1}{2}[\/latex] point. The candidate with the most points wins. <\/p>\n<p><strong>IIA Criterion:<\/strong> Removing a non-winning choice from the ballot should not change the election&#8217;s winner. <\/p>\n<p>Things to consider when applying Copeland&#8217;s method:<\/p>\n<ul>\n<li><strong>Pairwise Comparisons:<\/strong> Always compare candidates in pairs to determine the more preferred one.<\/li>\n<li><strong>Point Allocation:<\/strong> Award [latex]1[\/latex] point to the more preferred candidate and [latex]\\frac{1}{2}[\/latex] point to each in case of a tie.<\/li>\n<li><strong>Totaling Points:<\/strong> Sum up the points for each candidate; the one with the most points wins.<\/li>\n<li><strong>Check for IIA:<\/strong> Ensure that removing a non-winning candidate doesn&#8217;t change the election outcome.<\/li>\n<li><strong>Tie-Breaking:<\/strong> Copeland&#8217;s method can often result in ties, requiring more advanced methods for resolution.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Consider the preference schedule below, in which a company\u2019s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity. Determine the winner using Copeland\u2019s method.<\/p>\n<p>Initial votes :<\/p>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1st choice<\/td>\n<td>B<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>E<\/td>\n<\/tr>\n<tr>\n<td>2nd choice<\/td>\n<td>C<\/td>\n<td>A<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>E<\/td>\n<td>A<\/td>\n<\/tr>\n<tr>\n<td>3rd choice<\/td>\n<td>A<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<td>A<\/td>\n<td>A<\/td>\n<td>D<\/td>\n<\/tr>\n<tr>\n<td>4th choice<\/td>\n<td>D<\/td>\n<td>B<\/td>\n<td>A<\/td>\n<td>E<\/td>\n<td>C<\/td>\n<td>B<\/td>\n<\/tr>\n<tr>\n<td>5th choice<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>E<\/td>\n<td>B<\/td>\n<td>D<\/td>\n<td>C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q944614\">Show Solution<\/button> <\/p>\n<div id=\"q944614\" class=\"hidden-answer\" style=\"display: none\"> With [latex]5[\/latex] candidates, there are [latex]10[\/latex] comparisons to make:<\/p>\n<table>\n<tbody>\n<tr>\n<td>A vs B: [latex]11[\/latex] votes to [latex]9[\/latex] votes<\/td>\n<td>A gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>A vs C: [latex]3[\/latex] votes to [latex]17[\/latex] votes<\/td>\n<td>C gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>A vs D: [latex]10[\/latex] votes to [latex]10[\/latex] votes<\/td>\n<td>A gets [latex]\\frac{1}{2}[\/latex] point, D gets [latex]\\frac{1}{2}[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>A vs E: [latex]17[\/latex] votes to [latex]3[\/latex] votes<\/td>\n<td>A gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>B vs C: [latex]10[\/latex] votes to [latex]10[\/latex] votes<\/td>\n<td>B gets [latex]\\frac{1}{2}[\/latex] point, C gets [latex]\\frac{1}{2}[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>B vs D: [latex]9[\/latex] votes to [latex]11[\/latex] votes<\/td>\n<td>D gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>B vs E: [latex]13[\/latex] votes to [latex]7[\/latex] votes<\/td>\n<td>B gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>C vs D: [latex]9[\/latex] votes to [latex]11[\/latex] votes<\/td>\n<td>D gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>C vs E: [latex]17[\/latex] votes to [latex]3[\/latex] votes<\/td>\n<td>C gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<tr>\n<td>D vs E: [latex]17[\/latex] votes to [latex]3[\/latex] votes<\/td>\n<td>D gets [latex]1[\/latex] point<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Totaling these up: A gets [latex]2\\frac{1}{2}[\/latex] points B gets [latex]1\\frac{1}{2}[\/latex] points C gets [latex]2\\frac{1}{2}[\/latex] points D gets [latex]3\\frac{1}{2}[\/latex] points E gets [latex]0[\/latex] points. Using Copeland\u2019s Method, we declare D as the winner. Notice that in this case, D is not a Condorcet Winner. While Copeland\u2019s method will also select a Condorcet Candidate as the winner, the method still works in cases where there is no Condorcet Winner.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following for more information on the Copeland&#8217;s method.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/sh3IEALnpNg?si=7F5a0kyLtxii-VS7\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Copeland's+Method.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Copeland&#8217;s Method\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Approval Voting<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p><strong>Approval Voting:<\/strong> A voting method where you mark all choices you find acceptable. The option with the most approval wins. Approval voting can easily violate the majority criterion, meaning the candidate with the most first-choice votes may not win.<\/p>\n<p><strong>Strategic Insincere Voting:<\/strong> This method is susceptible to voters not voting their true preference to increase the chances of their choice winning.<\/p>\n<p>Things to consider when applying the approval voting method:<\/p>\n<ul>\n<li><strong>Marking Choices:<\/strong> In approval voting, you&#8217;re not ranking candidates; you&#8217;re simply marking all that you find acceptable.<\/li>\n<li><strong>Tallying Votes:<\/strong> Count the number of approvals each option receives. The one with the most wins.<\/li>\n<li><strong>Strategic Voting:<\/strong> Some voters may mark options they find less preferable to increase the chances of their true preference winning.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">A group of friends are trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an \u201cX\u201d which movies they are willing to watch. The results are:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">Bob<\/td>\n<td style=\"text-align: center;\">Ann<\/td>\n<td style=\"text-align: center;\">Marv<\/td>\n<td style=\"text-align: center;\">Alice<\/td>\n<td style=\"text-align: center;\">Eve<\/td>\n<td style=\"text-align: center;\">Omar<\/td>\n<td style=\"text-align: center;\">Lupe<\/td>\n<td style=\"text-align: center;\">Dave<\/td>\n<td style=\"text-align: center;\">Tish<\/td>\n<td style=\"text-align: center;\">Jim<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Avatar: The Way of Water<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Spider-Man: Across the Spider-Verse<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">Everything Everywhere All at Once<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">\u00a0<\/td>\n<td style=\"text-align: center;\">X<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q109152\">Solution<\/button> <\/p>\n<div id=\"q109152\" class=\"hidden-answer\" style=\"display: none\">\n<p>Totaling the results, we find:<\/p>\n<p>Avatar: The Way of Water received [latex]5[\/latex] approvals.<\/p>\n<p>Spider-Man: Across the Spider-Verse received [latex]6[\/latex] approvals.<\/p>\n<p>Everything Everywhere All at Once received [latex]7[\/latex] approvals.<\/p>\n<p>In this vote, Everything Everywhere All at Once would be the winner.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following for more information on the approval voting method.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vv1pquvAIDI?si=ZYJfXo-dVuna6u5l\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Approval+Voting.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Approval Voting\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Which Method is Fair?<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<\/strong><\/p>\n<p>Which Method is Fair? The search for a perfect voting method is elusive, as no method satisfies all fairness criteria.<\/p>\n<p><strong>Arrow&#8217;s Impossibility Theorem:<\/strong> Proven by Kenneth Arrow in 1949, this theorem states that no voting method can satisfy all fairness criteria.<\/p>\n<p><strong>Condorcet&#8217;s Voting Paradox:<\/strong> This paradox shows that voting preferences are not transitive, meaning that if A is preferred over B, and B over C, it doesn&#8217;t necessarily mean A is preferred over C.<\/p>\n<p><strong>Method Selection:<\/strong> The choice of voting method often depends on what seems most fair for the specific situation.<\/p>\n<p>Things to consider when evaluating voting methods and interpreting results:<\/p>\n<ul>\n<li><strong>Fairness Criteria:<\/strong> Always consider the fairness criteria when evaluating a voting method.<\/li>\n<li><strong>Transitivity:<\/strong> Be cautious of the Condorcet&#8217;s Voting Paradox when interpreting voting results.<\/li>\n<li><strong>Method Choice:<\/strong> The &#8220;best&#8221; method may vary depending on the context and what fairness criteria are most important.<\/li>\n<\/ul>\n<\/div>\n<p>Watch the following for more information on the four fairness criterion and the Arrow&#8217;s Impossibility Theorem.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/CuOLQT9P11I?si=I0cMmynZfoF7ZEbp\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Principles+of+Marketing+Transcriptions\/Voting+Theory_+Fairness+Criterion.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cVoting Theory: Fairness Criterion\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-3829-1\">This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See https:\/\/www.piercecountywa.gov\/DocumentCenter\/View\/6985\/summary?bidId= <a href=\"#return-footnote-3829-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-3829-2\">This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See https:\/\/www.piercecountywa.gov\/DocumentCenter\/View\/6985\/summary?bidId= <a href=\"#return-footnote-3829-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":15,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"(New Version Available) Introduction to Voting Theory and Preference Tables\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/6rhpq1ozmuQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Voting Theory: Plurality Method and Condorcet Criterion\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/r-VmxJQFMq8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Voting Theory: Borda count\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/hVG9jmA4FBU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Voting Theory: Copeland\\'s Method\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sh3IEALnpNg\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Voting Theory: Approval Voting\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/vv1pquvAIDI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Voting Theory: Fairness Criterion\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/CuOLQT9P11I\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":90,"module-header":"fresh_take","content_attributions":null,"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3829"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":41,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3829\/revisions"}],"predecessor-version":[{"id":14890,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3829\/revisions\/14890"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/90"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3829\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3829"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3829"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3829"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3829"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}